Hi Joy. The key to the answer to your question will be the details of your policy. In my opinion many trainers do NOT thoroughly read their policies and ask coverage questions until something bad has happened. So, I think that you’re being very smart to be proactive and ask this question now. What’s missing though is that you need to ask this question of your insurance company, not of this forum because ultimately it will be your insurer who will make the determination of what coverage you need.
Please, please, please contact your insurer and ask this very important question and do not rely on whatever answer you may receive here since none of us has read your actual policy.
I hope that this helps.
Check your policy with your insurance company. I teach Pilates on apparatus and recently became personal trainer certified. When I looked at my policy it covered both areas. You also want to make sure you buy the right amount of insurance for the number of hours you teach, i.e. 5 hours or less vs. more than 5 hours a week.